Hilbert–Arnold Problem

In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of limit cycles. It asks whether in a generic finite-parameter family of smooth vector fields on a sphere with a compact parameter base, the number of limit cycles is uniformly bounded across all parameter values. The problem is historically related to Hilbert's sixteenth problem and was first formulated by Russian mathematicians Vladimir Arnold and Yulij Ilyashenko in the 1980s.Ilyashenko, Yu. (1994). "Normal forms for local families and nonlocal bifurcations". ''Astérisque'', Vol. 222, 233-258.

It is closely related to the "infinitesimal Hilbert's sixteenth problem", although they are not synonyms. In ''Arnold's Problems'' there are many questions related to the Hilbert–Arnold problem: 1978–6, 1979–16, 1980–1, 1983–11, 1989–17, 1990–24, 1990–25, 1994–51 and 1994–52.

Intuitive, physical interpretation

The problem concerns the number of limit cycles of the dynamics of a flow on a sphere, and whether the number of such cycles can be bounded. A flow on a sphere means that you can imagine that the velocities of particles are prescribed, and a limit cycle is a limit of those velocities, like the Gulf Stream on the globeA similar interpretation can be found in Moore, A. M., & Mariano, A. J. (1999).
The Dynamics of Error Growth and Predictability in a Model of the Gulf Stream. Part I
: Singular Vector Analysis. ''Journal of Physical Oceanography'', 29(2), 158–176. https://doi.org/10.1175/1520-0485(1999)029<0158:tdoega>2.0.co;2 (free full access article) . The problem asks whether in a generic family of smooth vector fields, smoothly parameterized over a compact set in finite dimensional Euclidean space, the number of limit cycles is uniformly bounded across all parameter values. Thus, under perturbations of climate conditions, it asks whether there is a bounded number of "Gulf Streams".

Overview

The problem arises from considering modern approaches to Hilbert's sixteenth problem. While Hilbert's original question focused on polynomial vector fields, mathematical attention shifted toward properties of generic families within certain classes. Unlike polynomial systems, typical smooth systems on a sphere can have arbitrarily many hyperbolic limit cycles that persist under small perturbations. However, the question of uniform boundedness across parameter families remains meaningful and forms the basis of the Hilbert–Arnold problem.Ilyashenko, Yu.; Kaloshin, V. (1999). "Bifurcations of planar and spatial polycycles: Arnold's program and its development". Fields Inst. Commun., 24, 241-271.

Due to the compactness of both the parameter base and phase space, the Hilbert–Arnold problem can be reduced to a local problem studying bifurcations of special degenerate vector fields. This leads to the concept of polycycles— cyclically ordered sets of singular points connected by phase curve arcs—and their cyclicity, which measures the number of limit cycles born in bifurcations.

Local Hilbert–Arnold problem

The local version of the Hilbert–Arnold problem asks whether the maximum cyclicity of nontrivial polycycles in generic k-parameter families (known as the bifurcation number $B(k)$) is finite, and seeks explicit upper bounds.Kaloshin, V. (2001). "The Hilbert-Arnold problem and estimates for the cyclicity of polycycles on the plane and in space". ''Functional Analysis and Its Applications'', 35(2), 78–81.
The local Hilbert–Arnold problem has been solved for $k=1$ and $k=2$, with $B(1) = 1$ and $B(2) = 2$. For $k=3$, a solution strategy exists but remains incomplete. A simplified version considering only elementary polycycles (where all vertices are elementary singular points with at least one nonzero eigenvalue) has been more thoroughly studied. Ilyashenko and Yakovenko proved in 1995 that the elementary bifurcation number $E(k)$ is finite for all $k>0$.Ilyashenko, Yu.; Yakovenko, S. (1991). "Finitely-smooth normal forms of local families of diffeomorphisms and vector fields". '' Russian Mathematical Surveys'', 46(1), 3–39.

In 2003, mathematician Vadim Kaloshin established the explicit bound $E(k) < 25^$.Kaloshin, V. (2003). "The Existential Hilbert 16th Problem and an Estimate for Cyclicity of Elementary Polycycles". ''Inventiones Mathematicae'', 151, 451–512. https://arxiv.org/abs/math/0111053v1

See also

* Arnold's_Problems#Notable_problems
*Bifurcation theory
*Limit cycle
*List of unsolved problems in mathematics

References

{{reflist

Further reading

* Yu. Ilyashenko.
Centennial History of Hilbert's 16 Problem
, Bulletin of the AMS, Volume 39, Number 3, pp. 301–354
* Bravo, J. L., Mardešić, P., Novikov, D., & Pontigo-Herrera, J. (2025). "Infinitesimal and tangential 16-th Hilbert problem on zero-cycles". ''Bulletin Des Sciences Mathématiques'', 202, 103634. https://doi.org/10.1016/j.bulsci.2025.103634 https://arxiv.org/abs/2312.03081
*V. Arnold, Yu. Ilyashenko, "Ordinary differential equations, Dynamical systems–1", ''Encyclopedia of Mathematical Sciences'', vol. 1, Springer, 1988, pp. 1–148.

Dynamical systems
Systems theory
Unsolved problems in mathematics
Bifurcation theory